TITLE: Geometrical Meaning of derivative of complex function
QUESTION [15 upvotes]: What's the geometrical meaning of $f'(z)$ in complex analysis, as we know in real analysis $f'(x)$ has meaning ie. Slope of curve or gives max/ min. But what does derivative $f'(z)$ has geometrical meaning in complex analysis

REPLY [30 votes]: Multiplying by a complex number other than $0$ consists of rotating and dilating.  To multiply by $i$ is to rotate $90^\circ$ counterclockwise; to multiply by $4+3i$ is to rotate counterclockwise through the angle $\arctan(3/4)$ and dilate by $5=\sqrt{3^2+4^2}$, etc.
So say we have
$$
\left.\frac{dw}{dz}\right|_{z=z_0} = f'(z_0). 
$$
Then at $z=z_0$ we have $dw=f'(z_0)\,dz$, i.e. if $dz$ is an infinitely small change in $z$, from $z_0$ to $z_0+dz$, then the corresponding infinitely small change in $w$ from $f(z_0)$ to $f(z_0) + dw$, is what you get from rotating and dilating $dz$ by the amounts corresonding to $f'(z_0)$.
This explains why holomorphic functions are conformal except at points where the derivative is $0$: where two curves intersect, the process of rotating does not change the angle between them since they're both rotated by the same amount, and of course dilating does not change the angle.